Harry owes the bank money. To repay his debt, he paid $\$150$ back to the bank each month. After $10$ months, his remaining debt was $\$6900$. What was the original amount of Harry's debt? $\$$
Harry paid $\$150$ back to the bank each month, so he paid $150T$ dollars in $T$ months. Harry's remaining debt is found by taking the original debt and subtracting from it the money he already paid back. We can express this with the equation $R=L-150T$, where: $R$ represents Harry's remaining debt at a given time (in dollars) $L$ represents the original debt (in dollars) $T$ represents the time (in months) We want to find $L$, so let's first solve the equation for $L$ : $ \begin{aligned}R&=L-150T\\ L&=R+150T\end{aligned}$ Now, we know that after $10$ months $(T={10})$, Harry's remaining debt is $\$6900$ $(R={6900})$. Let's plug these values into the equation to find the value of $L$. $ L={6900}+150\cdot{10}=8400$ Therefore, Harry's original debt was $\$8400$. To find how long it took Harry to pay back his entire debt, we can plug $R=0$ into the equation and solve for $T$. $ \begin{aligned}8400&=0+150T\\ 150T&=8400\\ T&=56\end{aligned}$ Harry's original debt was $\$8400$. It took Harry $56$ months to pay back his entire debt.